Four wires of the same diameter are connected, in turn, between two points maintained at a constant potential difference, Their resistivities and lengths are, `rho and L ("wire 1") 1.2rho and 1.2L("wire 2"), 0.9 rho and 0.9L("wire 3") and rho and 1.5L ("wire 4")`. Rank the wires according to hte rates at which energy is dissipated as heat, greatest first,

Options

A. `4 gt 3 gt 1 gt 2 `
B. `4 gt 2 gt 1 gt 3 `
C. `1 gt 2 gt 3 gt 4`
D. `3 gt 1 gt 2 gt 4`

Explanation

To solve the problem of ranking the wires according to the rates at which energy is dissipated as heat, we will follow these steps:

### Step 1: Calculate the Resistance of Each Wire
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity, \( L \) is the length, and \( A \) is the cross-sectional area of the wire. Since all wires have the same diameter, the area \( A \) will be constant for all wires.

1. **Wire 1:**
\[ R_1 = \frac{\rho L}{A} \]

2. **Wire 2:**
\[ R_2 = \frac{1.2\rho \cdot 1.2L}{A} = \frac{1.44\rho L}{A} \]

3. **Wire 3:**
\[ R_3 = \frac{0.9\rho \cdot 0.9L}{A} = \frac{0.81\rho L}{A} \]

4. **Wire 4:**
\[ R_4 = \frac{\rho \cdot 1.5L}{A} = \frac{1.5\rho L}{A} \]

### Step 2: List the Resistance Values
Now we have the resistance values for each wire:
- \( R_1 = \frac{\rho L}{A} \)
- \( R_2 = \frac{1.44\rho L}{A} \)
- \( R_3 = \frac{0.81\rho L}{A} \)
- \( R_4 = \frac{1.5\rho L}{A} \)

### Step 3: Determine the Heat Dissipation Rate
The rate of heat dissipation \( H \) in a resistor when a constant potential difference \( V \) is applied is given by:
\[ H = \frac{V^2}{R} \]
Since \( V \) is constant for all wires, we can say that the heat dissipation is inversely proportional to the resistance:
\[ H \propto \frac{1}{R} \]

### Step 4: Rank the Wires by Resistance
Now we rank the wires based on their resistance values:
- \( R_3 < R_1 < R_2 < R_4 \)

### Step 5: Rank the Wires by Heat Dissipation Rate
Since heat dissipation is inversely proportional to resistance, we can reverse the order:
- \( H_3 > H_1 > H_2 > H_4 \)

### Final Ranking
Thus, the final ranking of the wires according to the rates at which energy is dissipated as heat, from greatest to least, is:
1. Wire 3
2. Wire 1
3. Wire 2
4. Wire 4